Nuprl Lemma : reg-seq-list-add_functionality_wrt_bdd-diff
∀L,L':(ℕ+ ⟶ ℤ) List.
(((||L|| = ||L'|| ∈ ℤ) ∧ (∀i:ℕ||L||. bdd-diff(L[i];L'[i]))) ⇒ bdd-diff(reg-seq-list-add(L);reg-seq-list-add(L')))
Proof
Definitions occuring in Statement :
reg-seq-list-add: reg-seq-list-add(L),
bdd-diff: bdd-diff(f;g),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
so_apply: x[s],
map: map(f;as),
list_ind: list_ind,
nil: [],
it: ⋅,
select: L[n],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
ge: i ≥ j ,
le: A ≤ B,
uiff: uiff(P;Q),
subtract: n - m,
less_than: a < b,
less_than': less_than'(a;b),
nat_plus: ℕ+,
cons: [a / b],
bdd-diff: bdd-diff(f;g),
nat: ℕ,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
bdd-diff_wf,
squash_wf,
true_wf,
nat_plus_wf,
reg-seq-list-add-as-l_sum,
iff_weakening_equal,
equal_wf,
length_wf,
all_wf,
int_seg_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
list_wf,
list_induction,
l_sum_wf,
map_wf,
equal-wf-base-T,
nil_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
map_nil_lemma,
l_sum_nil_lemma,
equal-wf-base,
length_of_cons_lemma,
map_cons_lemma,
l_sum_cons_lemma,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
cons_wf,
add-is-int-iff,
false_wf,
equal-wf-T-base,
decidable__equal_int,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
lelt_wf,
select-cons-tl,
add-subtract-cancel,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_properties,
nat_properties,
le_wf,
absval_wf,
trivial-bdd-diff,
nat_wf,
itermMinus_wf,
int_term_value_minus_lemma,
and_wf,
le_functionality,
le_weakening,
int-triangle-inequality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
applyEquality,
lambdaEquality,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
functionEquality,
intEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
productEquality,
because_Cache,
setElimination,
rename,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
functionExtensionality,
addEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}L,L':(\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}) List.
(((||L|| = ||L'||) \mwedge{} (\mforall{}i:\mBbbN{}||L||. bdd-diff(L[i];L'[i])))
{}\mRightarrow{} bdd-diff(reg-seq-list-add(L);reg-seq-list-add(L')))
Date html generated:
2017_10_02-PM-07_14_01
Last ObjectModification:
2017_07_28-AM-07_20_09
Theory : reals
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